# Questions tagged [galois-representations]

The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.

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### Power series ring $\Theta[[X_1,\ldots,X_d]]$ and prime ideals

Let $\Theta$ be a domain. We shall choose $d$ elements $\theta_1,\ldots,\theta_d \in \Theta$ such that any chosen $j$ elements $\theta_{i_1},\ldots,\theta_{i_j}$ form a prime ideal $(\theta_{i_1},\...

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### Stacks project for Galois representations and automorphic forms

Is there anything like Stacks project for Galois representations and automorphic forms? I am not asking people to start something like Stacks project, just asking if something like Stacks project ...

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### Level Lowering Galois representations over Totally real fields

Let $F$ be a totally real number field and $\mathbb{F}$ a finite field. Let $\bar{\rho}:\text{Gal}(\bar{F}/F)\rightarrow \text{GL}_2(\mathbb{F})$ an irreducible Galois representation arising from a ...

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### Derived weight filtration on motivic Galois representations

Thanks to modern techniques (such as the pro-etale site), we can now understand etale (co)homology of varieties and motives as "genuinely" derived (e.g. DG) Galois-equivariant objects. I'm looking for ...

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### Fontaine-Fargues curve and period rings and untilt

When I read the paper "THE FARGUES–FONTAINE CURVE AND DIAMONDS" of Matthew Morrow, I have a question on page 11.
Question: The arthur said that the de Rham and crystalline period rings implicitly ...

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### How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?

Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...

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### Classify 2-dim p-adic galois representations

Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...

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### Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM

Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Gr?ssencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$...

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### Comparison of weight-monodromy filtrations

Setup:
Let $R$ be a finitely generated subring of $\mathbb{C}$. Let $X \rightarrow \mathbb{A}^1_R$ be a proper morphism of $R$-varieties, smooth except over a rational point $s \in \mathbb{A}^1_R$ ...

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### Classification of finite flat group schemes over integers?

One can classify (commutative) finite flat group schemes (with order of $p$-powers) over $\mathbb Z_p$ using semi-linear algebraic datas such as Breuil-Kisin modules. And we can fix the special fiber ...

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### Reference for Selmer-Group coming from Galois representation associated with modular form

Is there any good reference (lecture notes) for the construction of Selmer-Groups associated with the Galois representation? In particular, i want to understand how they are using Deligne's and Mazur'...

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### On the product in the power series ring

Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...

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### Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group

In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said
"We say that
$\rho$ is crystalline/de Rham/Hodge–Tate if ...

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### Lifting Galois representations

Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}$ or $\mathbb{Q}_p$. Let $k$ be a perfect field of characteristic $l>0$ (possibly $l=p$).
If we have a homomorphism $G\...

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### Hilbert modular form as a representation of Hecke algebra

I am reading some notes by Snowden and I don't understand a sentence.
Clearly, if we have an appropriate $R = T$
theorem then we get a modularity lifting theorem, as $\rho$ defines a homomorphism ...